Two coils have self-inductance `L_(1) = 4mH` and `L_(2) = 1 mH` respectively. The currents in the two coils are increased at the same rate. At a certain instant of time both coils are given the same power. If `I_(1)` and `I_(2)` are the currents in the two coils, at that instant of time respectively, then the value of `(I_(1)//I_(2))` is :

Options

A. `(1)/(8)`
B. `(1)/(4)`
C. `(1)/(2)`
D. 1

Explanation

To solve the problem, we need to find the ratio of the currents \( I_1 \) and \( I_2 \) in two coils with given self-inductances, under the condition that they are supplied with the same power. Let's go through the solution step by step.

### Step 1: Write the expressions for the induced EMF
The induced EMF (voltage) across each coil can be expressed using the formula for self-inductance:
\[ V_1 = L_1 \frac{dI_1}{dt} \]
\[ V_2 = L_2 \frac{dI_2}{dt} \]
where \( L_1 = 4 \, \text{mH} \) and \( L_2 = 1 \, \text{mH} \).

### Step 2: Write the power equations
The power \( P \) in each coil can be expressed as:
\[ P_1 = V_1 I_1 \]
\[ P_2 = V_2 I_2 \]
Since both coils are given the same power at a certain instant, we can equate the two power expressions:
\[ V_1 I_1 = V_2 I_2 \]

### Step 3: Substitute the expressions for voltage
Substituting the expressions for \( V_1 \) and \( V_2 \) into the power equation gives:
\[ (L_1 \frac{dI_1}{dt}) I_1 = (L_2 \frac{dI_2}{dt}) I_2 \]

### Step 4: Substitute the values of self-inductance
Substituting \( L_1 = 4 \, \text{mH} \) and \( L_2 = 1 \, \text{mH} \):
\[ (4 \times 10^{-3} \frac{dI_1}{dt}) I_1 = (1 \times 10^{-3} \frac{dI_2}{dt}) I_2 \]

### Step 5: Simplify the equation
This simplifies to:
\[ 4 I_1 \frac{dI_1}{dt} = I_2 \frac{dI_2}{dt} \]

### Step 6: Use the condition of equal rates of change of current
Since the currents in both coils are increased at the same rate, we have:
\[ \frac{dI_1}{dt} = \frac{dI_2}{dt} \]
Thus, we can cancel \( \frac{dI_1}{dt} \) and \( \frac{dI_2}{dt} \) from both sides:
\[ 4 I_1 = I_2 \]

### Step 7: Find the ratio of the currents
Rearranging gives:
\[ \frac{I_1}{I_2} = \frac{1}{4} \]

### Conclusion
Thus, the value of \( \frac{I_1}{I_2} \) is:
\[ \frac{I_1}{I_2} = \frac{1}{4} \]